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Russian Microelectronics

A Quantum-Search-Based Algorithm for the 0-1 Knapsack Problem

K. A. Stepanenko

Published Dec 1, 2025• 0 citations
Publisher

Abstract

Abstract The knapsack problem is an important example of a combinatorial optimization problem with a wide range of applications in direct practical applications. In this article, the author considers the classical version of the problem (0-1 knapsack) and proposes a quantum algorithm for solving it based on a hybrid approach that combines the Grover quantum search algorithm with the procedure for finding the Durr–Hoyer minimum. The paper presents an analysis of the computational complexity of the proposed algorithm, and in addition to theoretical analysis, the article contains an explicit description of the quantum circuit implementing the proposed algorithm.

DOI
10.1134/s1063739725602000

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Abstract Solving combinatorial optimization problems represents a challenge for researchers, especially hard NP-complete problems that are commonly handled through heuristic methods providing approximate solutions. The emergence of quantum computing, which enables performing parallel computation, raises hope for solving such problems within a reasonable time frame. This paper proposes a quantum global optimization algorithm for solving combinatorial optimization problems with improved performance and runtime approaching Grover’s optimal time of quantum search. The algorithm determines the global optimum of the objective function defining the addressed problem using an enhanced application of Grover’s search by simultaneously exploring all comparison cases of possible solutions without requiring iterative operations, unlike previous approaches, which affect performance and runtime. The paper also introduces a version of the proposed algorithm for solving the knapsack problem, along with its execution results and a comparison with other competing algorithms.
Abstract Solving combinatorial optimization problems represents a challenge for researchers, especially hard NP-complete problems that are commonly handled through heuristic methods providing approximate solutions. The emergence of quantum computing, which enables performing parallel computation, raises hope for solving such problems within a reasonable time frame. This paper proposes a quantum global optimization algorithm to solve combinatorial optimization problems intended to provide an improved performance. The proposed algorithm aims to amplify the probability of selecting the global optimum of the objective function defining the addressed problem using a specific variant of Grover’s search based on simultaneously exploring all comparison cases of possible solutions to recognize the global optimum. The paper also includes the study of precision and runtime of the presented concept and introduces a version of the proposed algorithm addressing the knapsack problem, along with its execution results and a comparison with other competing algorithms
Abstract Solving combinatorial optimization problems represents a challenge for researchers, especially hard NP-complete problems that are commonly handled through heuristic methods providing approximate solutions. The emergence of quantum computing, which enables performing parallel computation, raises hope for solving such problems within a reasonable time frame. This paper proposes a quantum global optimization algorithm to solve combinatorial optimization problems intended to provide an improved performance. The proposed algorithm aims to amplify the probability of selecting the global optimum of the objective function defining the addressed problem using a specific variant of Grover’s search based on simultaneously exploring all comparison cases of possible solutions to recognize the right one. The paper also includes the study of precision and runtime of the presented concept and introduces a version of the proposed algorithm addressing the knapsack problem, along with its execution results and a comparison with other competing algorithms.
The Multi-Dimensional Knapsack Problem (MDKP) is a strongly NP-Hard combinatorial optimization problem that extends the Knapsack Problem (KP) to multiple constraints. While effective heuristic classical MDKP algorithms have been studied, there remains no optimal exact classical algorithm and only few contributions using quantum algorithms exist. In this study, we leverage Grover's search and pre-processing techniques to propose an iterative quantum search algorithm and circuit design that integrates classical adjustments to the oracle, progressively refining solutions until optimal. Evaluations show that our method reduces depth, width, iterations, and runtime.
The Knapsack Problem is a prominent problem that is used in resource allocation and cryptography. This paper presents an oracle and a circuit design that verifies solutions to the decision problem form of the Bounded Knapsack Problem. This oracle can be used by Grover Search to solve the optimization problem form of the Bounded Knapsack Problem. This algorithm leverages the quadratic speed-up offered by Grover Search to achieve a quantum algorithm for the Knapsack Problem that shows improvement with regard to classical algorithms. The quantum circuits were designed using the Microsoft Q# Programming Language and verified on its local quantum simulator. The paper also provides analyses of the complexity and gate cost of the proposed oracle. The work in this paper is the first such proposed method for the Knapsack Optimization Problem.
Combinatorial optimization problems play a vital role in computer science and engineering, with applications in logistics, network design, finance, and resource allocation. However, many of these problems are NP-hard, making them computationally expensive to solve using classical optimization techniques, especially for large-scale instances. In recent years, quantum computing has emerged as a promising paradigm capable of addressing these limitations by leveraging quantum mechanical principles such as superposition and entanglement. This study investigates the application of quantum computing for solving combinatorial optimization problems, with a focus on algorithm design, complexity analysis, and real-world applicability. The research primarily employs the Quantum Approximate Optimization Algorithm (QAOA) and Grover's search algorithm to solve representative problems such as Max-Cut and the Knapsack problem. The performance of these algorithms is evaluated using key metrics including accuracy, execution time, and approximation ratio, and is compared with classical optimization techniques. The results demonstrate that QAOA is capable of generating near-optimal solutions, achieving accuracy levels of up to 93% for small problem instances. However, the study also highlights the challenges associated with current quantum systems, including noise, limited qubit availability, and increased execution time. Comparative analysis reveals that while classical methods remain more efficient for small-scale problems, quantum algorithms show significant potential for scalability and handling complex optimization tasks. The findings suggest that hybrid quantum-classical approaches offer a practical solution in the current Noisy Intermediate-Scale Quantum (NISQ) era. The study concludes that although quantum computing is still evolving, it holds strong potential for transforming combinatorial optimization in the future.